On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F = F-q(t) having conductor (p)(.)infinity, where (p) is a prime ideal in F-q[t]. Let delta is an element of F-q[t] be an irreducible polynomial of odd degree, and let K = F(root delta). Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E circle times(F)K, 1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group III(E/K) when L(E circle times(F)K, 1) not equal 0. (c) 2005 Elsevier Inc. All rights reserved.